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Appendix A: From images to complex visibilities

The radiative-transfer code MCMax produces images on a predefined grid of pixels on the sky. We chose a 256 × 256 grid covering a field-of-view of 100 mas × 100 mas. This results in an angular resolution of 0.39 mas per pixel.

The model image produced by MCMax for a given set of parameters (M_⋆  ⋆, R_in, R_out, M_dust, p, i) has a default disk orientation angle on the sky of 0° E of N. That is, the far side of the inclined disk is oriented towards the north. This image is rotated according to the grid disk PA under consideration to yield the model image corresponding to (M_⋆  ⋆, R_in, R_out, M_dust, p, i, PA).

For a specific observational setting with baseline length b, baseline orientation angle θ, and at wavelength λ, the van Cittert-Zernicke theorem gives the complex normalized visibility (A.1)where (u = bsinθ/λ,v = bcosθ/λ) are the spatial frequencies, (α,β) the angular coordinates, and B(α,   β) the brightness distribution (i.e., the image) of the source on the sky, normalized to a total intensity of unity.

To compute , the image is rotated through the angle θ. This is equivalent to a coordinate transform such that (u′,v′) = (0,b/λ) and (α′,β′) = (αcosθ − βsinθ,αsinθ + βcosθ). Equation (A.1) becomes Numerically, the integration over α′ reduces to a sum over the first dimension of the rotated image B(α′,β′). The second integration is a one-dimensional Fourier transform and can be done quickly with the fast Fourier transform (FFT) algorithm. However, FFT converts the 256-array with a pixel step of 0.39 mas into a 256-array with a spatial-frequency step of 10 cycles/arcsec. Since the observations are obtained at spatial frequencies in the range 25−370 cycles/arcsec, the default FFT resolution is too coarse for direct comparison. To increase the resolution in spatial frequency, the 256-array (i.e., ^∫dα′B(α′,β′)) is placed in a 1024-array, which is zero outside the image. The FFT of the latter array yields a spatial-frequency step of 2.5 cycles/arcsec. This higher resolution makes the FFT result smooth enough to allow interpolation at the exact spatial frequencies of the observations.

The method was tested on images of simple geometries (point source, binary with unresolved components, uniform disk, uniform ellipse), for which analytic formulae of the complex visibility exist. Good agreement was found. For a uniform disk with a diameter of 10 mas, the absolute difference between the analytic and image-based squared visibilities is 0.002 on average, and always below 0.01. The absolute difference between the analytic closure phase (i.e., 0° or 180° for a uniform disk) and the image closure phase is 1° on average, but can increase to 10° at the longest baselines. At these high spatial frequencies, pixelation effects start to play a role for the closure phases, while they do not affect the visibilities to a measurable level⁹.

The obtained accuracy of the model is sufficient for the goals of this paper. An increase of the spatial resolution of the model image is possible, but would lead to an increase of the grid computation time from days to weeks.

Appendix B: Stellar contribution in the near-infrared

In the near-IR, the flux contribution of the central star to the total flux can be large. The stellar flux contribution of the primary of HR 4049 drops from ~0.6 at 1.6 μm to ~0.2 at 2.5 μm. The intensity-normalized image of the disk, on the other hand, is similar in the H and K band. We therefore took the following approach to compute squared visibilities and closure phases in the near-IR:

• 1.

  From the radiative-transfer model, we computed a2.2 μm image. The central2 × 2 pixels in this image contain the primary star. We removed the star from the image by replacing these central pixel values by interpolated values. Hence, we created a star-less image, which provides a representative disk image at all wavelengths between 1.6 and 2.5 μm.

• 2.

  We computed the disk complex visibilities from the star-less image as described in Appendix A.

• 3.

  From the model infrared spectrum and the input Kurucz model for the primary star, the stellar-to-total flux contribution f was estimated at all AMBER wavelengths.

• 4.

  The final model visibilities were computed from the disk visibilities and the flux ratio f, according to (B.1)From this complex quantity, squared visibilities (per baseline and wavelength) and closure phases (per baseline triangle and wavelength) were computed. Given its angular diameter (0.45 ± 0.03 mas, determined from the Kurucz model fit to the optical photometry), the primary star is unresolved at the spatial resolution of the observations. Moreover, we assumed the primary star is at the disk’s center and hence . Note that we neglected the possible offset of the primary star with respect to the disk center due to binary motion. However, given the dimensions of the disk (~15 mas) and of the binary (~1 mas), the effect of such an offset on visibilities and closure phases is expected to be small. Moreover, the interferometric observations were taken close to minimal brightness, when the primary star was close to the observer. The primary was therefore close to projected minor axis of the disk and close to the disk center on the sky.

In the mid-IR, the relative flux contribution of the primary star to the total flux is less than a few percent. Moreover, it is constant over the N band because of the near-blackbody shape of both the stellar flux and the excess. The intensity-normalized model image of HR 4049 at 8 μm is therefore the same as at 13 μm. The variations in (relative) visibility over the N band are exclusively due to a change in spatial frequency and not due to a change in disk geometry or stellar flux contribution with wavelength.

© ESO, 2013